Ball-drop from a tall building (à la Galileo)

We explore the physics of falling bodies by dropping various balls from the 15th floor (estimated to be about 45 to 50 metres) of a building and compare it with drops from lower heights. Check this article and the associated video first, as we discuss the physics behind that activity here.

When an object such as a tennis ball is dropped from a height, the collision it makes with the ground or floor may be inelastic or elastic to various degrees. If it is perfectly elastic, we expect it to bounce to the same height owing to energy conservation. Otherwise the restitution coefficient determines how far back it bounces to. But is the restitution coefficient a constant, and in what régime? In other words, should bounce-back fraction of height not be dependent on the height you drop it from?

A bunch of us took a sampling of balls such as basketball, tennis ball, a steel bob to test this: on dropping them to the floor in a room from heights of the order of a metre, the tennis ball bounced back to about 23 of its drop height. The steel ball actually didn’t do that well (even though we all know that steel is more elastic than rubber! Interestingly, the steel bob would bounce back higher than a rubber ball when the collision is against another steel bob although it’s a bit harder to aim…but think of Newton’s Cradle). The basketball’s elasticity stems, to a large extent, from the air pressure inside as we all know.

So if you drop these balls onto tiled ground from the 15th floor of a tall building, how high would they bounce? Check the video!

So they all bounce to no greater than the 3rd or the 4th floor whereas we expect it to go to the 10th floor (steel not tested so as not to break infrastructure!). To explain this anomaly, one could model the restitution for normal drop tests for various materials of the ball (such as this work pointed out by Nandita). Let’s take a look at the individual physical processes: we shall discuss two important physical ideas behind the lacklustre performance of the ball.

One is the air viscosity, and check if the reason is the same as why raindrops don’t fall with bullet speeds. The air around the ball could be said to undergo Stokes’ flow, i.e. laminar, if the dimensionless Reynolds’ number is small (say, less than unity). Given enough time, the ball may attain a sizeable fraction of its terminal velocity. Let the radius of the tennis ball be 3.5 , its mass 60 , and the viscosity of air be 1E‑5 at a density of 1.2 . Though it is a spherical shell and hence slightly different, we’ll use an average density and go ahead with Stokes’ analysis for a uniform sphere (an interesting question would be if that’s ok?). We actually require just the mass of the ball, and because of the large density ratio between the ball and air we may neglect buoyancy too. We obtain using above values, the time required to reach 95% of terminal velocity (equivalently, to be within a respectable 5% error margin) or 90000 as 27000 . Newtonian kinematic equations suggest a more realistic time spent in free fall as 3 during which it gains a velocity of about 30 (calculations courtesy Jayanth Vyasanakere).

The Reynolds number from this simplistic analysis is also huge: a billion! Obviously then, the viscous drag on the ball cannot be laminar, so the flow of air around it might be turbulent. Even without detailed calculation, one notices the immediate effect of turbulence with the drag being proportional to the square of the velocity instead of being linear as in laminar flow.

Another contributing factor could be related to what is a serious issue in the oil and natural gas industry, known as subsidence. Quite simply the ground underneath may have sunk a bit upon impact, thereby taking some of the available kinetic energy (thereafter turning it to the motion of sand particles, heat etc.) and making the collision a lot more inelastic than one would imagine. Probably the same as the steel bob on floor tile? The basketball, as pointed out by Alahi in the video, even made a spiral!

It’ll probably take much more effort to list down the factors to model the ball drop to any degree of higher precision.